How to prove that $n(\log_3(n))^5 = O(n^{1.2})$?

Question

This a homework question from Udi Manber's book. Any hint would be nice :)

I must show that:

$n(\log_3(n))^5 = O(n^{1.2})$

I tried using Theorem 3.1 of book:

$f(n)^c = O(a^{f(n)})$ (for $c > 0$, $a > 1$)

Substituing:

$(\log_3(n))^5 = O(3^{\log_3(n)}) = O(n) $

but $n(\log_3(n))^5 = O(n\cdot n) = O(n^2) \ne O(n^{1.2})$

Thank you for any help.

What methods can you use? take a look at this answer it might give you some ideas. Also here there is plenty of useful information. — Ran G., Apr 02 '12 at 01:10
@RanG. should this be closed in the light of the linked question — Suresh, Apr 02 '12 at 01:13
@Suresh I'm not sure. I fear if we don't we would be flooded with such questions (which maybe should fit [math.se] better). But it is a valid question. — Ran G., Apr 02 '12 at 01:15
@RanG.: math.SE is already flooded with these questions, mostly tagged "algorithms". — Louis, Apr 02 '12 at 11:26

score 14 · Accepted Answer · edited May 24 '13 at 03:24

14

Do what you did, but let $a = (3^{0.2})$... that should do it, right?

The reason that what you did didn't work is as follows. The big-oh bound is not tight; while the logarithm to the fifth is indeed big-oh of linear functions, it is also big oh of the fifth root function. You need this stronger result (which you can also get from the theorem) to do what you're doing.

edited May 24 '13 at 03:24

Juho

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answered Apr 02 '12 at 01:17

Patrick87

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In fact, for any $\epsilon >0$, $n \log^c n = O(n^{1+\epsilon})$ – Ran G. Apr 02 '12 at 01:28
@RanG. Yes, that is a direct consequence of the theorem. – Patrick87 Apr 02 '12 at 03:16
@AndreResende If my answer helped you solve your problem, and it makes sense, you can "accept" using the green check mark. It helps others see what worked for you, and might help you get more help in the future. Of course, if you'd like other answers, hold out. – Patrick87 Apr 02 '12 at 03:19

score 5 · Answer 2 · answered Apr 02 '12 at 04:14

Another way to think about it more intuitively, is to see that the main thing you have to show is that $(\log_3(n))^5$ is $O(n^{0.2})$, or equivalently that $\log_3(n)$ is $O(n^{0.04})$. Logs always grow slower than any constant power of n, no matter how small.

If you want to formalize the last sentence, then you can use theorem 3 with a sufficiently small $\alpha$ as @RanG mentions in the comment on @Patrick87's answer.

How to prove that $n(\log_3(n))^5 = O(n^{1.2})$?

2 Answers2